Most hatchery Easter Eggers originated from the first "mutts" that laid colored eggs who were brought over from South America, most of them actually being the N. American Quechua though. Through time they've either been crossbred with production layers or kept the same, it varies, really. The most "pure" and original blood being from McMurray hatchery.
Remember, today's Ameraucana and Araucana were developed from varied colored-egg laying mutts from South American and Central America.
Otherwise, any other non-hatchery "homemade" Easter Egger is any bird who's parent(s) carry at least one blue egg gene, yet the offspring does not meet the Standard for Ameraucanas or Araucanas. So, you can have an EE x Polish, or an Araucana x Marans, or an Ameraucana x EE, or whatever - And the result is an Easter Egger
The blue egg gene evidently originated in South America, Chile probably. There are several different recognized breeds worldwide that have the blue egg gene. In the US the only two that are officially recognized are the Araucana and Ameraucana. The criteria for specific breeds can be different on different countries, so the criteria for lets say the Araucanas might be different in Europe than in the US. And in Chile, the Araucana are probably more of a type than a recognized breed.
Many people in the US link the EE with the Ameraucana. Somewhere along the line, someone took some chickens that had the blue egg gene and developed the Ameraucana and Araucana breed, including not only the blue egg gene but leg color, certain colors and patterns, comb types, configutation, skin color, and many other things. A lot of people think an EE should look like an Ameraucana, though an EE could have absolutely no link to an Ameraucana. They may be linked to the Araucana or something totally different.
They are considered mutts because there are no officially recognized standards for an EE against which they can be judged.